Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Naturality of Lie derivatives

It was left to me as an exercise to show the naturality of the Lie derivative of arbitrary tensors on a smooth manifold. Is the following argument correct (it seems too easy)? Let $X$ be a vector ...
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3answers
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The principal curvatures of a surface of revolution

The principal curvatures of the surface at a point is defined as the maximal and the minimal curvature among all normal sections. It's claimed (say, on Stillwell's Geometry of Surfaces) that for a ...
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15 views

Orthogonal transformation and vector product

I found these thing in an exercise 1.5.6 in the book Differential Geometry of curves and surfaces - Do Carmo. "Show that the vector product of 2 vectors is invariant under orthogonal transformation ...
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1answer
2 views

Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
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1answer
26 views

Degree of a vector field on the boundary of a punctured ball?

Suppose $v\colon \mathbb{R}^3\setminus\{x_1,\dots,x_n\}\to S^2$ is a smooth map, where all the points $x_i$ sit inside the unit ball $B^3$, and not on the boundary $S^2$. I suppose this map can be ...
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1answer
20 views

Finding the surface area of a parametrized surface

I was wondering how you would compute the surface area of a parameterized surface. Is there a formula or set of procedures you can follow to compute this. Say I wanted to compute the surface area of a ...
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21 views

When to take derivative with respect to distance?

I had a previous question about the divergence in spherical coordinates and using the usual formula found on wikipedia "List of formulas in Riemannian geometry" I could not get the correct form of the ...
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19 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
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13 views

Implicit representations of a regular surface.

Suppose that $\mathcal{S}$ is a regular surface and $f(x,y,z)=0$ is an implicit representation of this surface in a neighbourhood $V$. Can it be shown in general that at any point of $V$ one of ...
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11 views

Relationship between differentiation and integration of vector fields?

Let $V\in\Gamma(T\mathbb{R}^n)$ be a vector field and $\gamma:[a,b]\to \mathbb{R}^n$ a curve. Let $\nabla$ be the Euclidean connection, i.e. $\nabla_XY=XY^k\frac{\partial}{\partial x^k}$. We have a ...
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219 views

How do you imagine the shape of a manifold $S^2 \times S^1$?

In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times. (The following story can be applied not only this manifold but also for any 3-dimensional manifold.) But ...
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1answer
37 views

Why are parallel vector fields called parallel?

In Lee's "Riemannian Manifolds: An Introduction to Curvature" given a curve $\gamma:[a,b]\to M$ and a tangent vector $V_0\in T_{\gamma(t_0)}M$, where $t_0\in [a,b]$, there is a drawing of the parallel ...
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2answers
64 views
+50

Functional of Einstein tensor

What does this equal to, and how do I actually calculate this correctly? $$ \frac{\delta G_{ab}}{\delta g_{cd}}=? $$
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1answer
61 views

Confusion regarding Riemann normal coordinates

I'm trying to understand Riemann normal coordinates. This "simple" example using the surface of a unit sphere is from http://www.maths.bris.ac.uk/~macpd/gen_rel/snotes.pdf (p26). The “north pole” ...
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1answer
23 views

Real/Complex Manifolds - Transition Maps

I'm trying to understand how real/complex structure is imposed on a manifold, especially the likes of smooth manifolds. I can read the definitions and work with them, but I want to understand ...
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1answer
427 views

exact differential n-forms

We know that a 1-form $\omega$ on a manifold $M$ is exact if and only if $\int_{\gamma}\omega=0$ for any closed loop $\gamma$. How can I prove the following generalization: $\omega$ is an exact n-form ...
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23 views

Computing transition map of $S^2$.

First please have a look at the cruddy diagram I have drawn. (it is at angle because my camera casts a shadow if I photograph it from above) Define the coordinate charts that map a portion of the ...
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35 views

Cohomology class with trivial restriction to a very general fiber

Let $f:X\to S$ be a flat morphism of smooth complex projective varieties. Let $s\in S$ be a very general point. Suppose that $\omega\in H^{p,p}(X)$ is a cohomology class such that $\omega|_{X_s}=0\in ...
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1answer
45 views

Sobolev space of p-forms on a Riemannian mamifold

Let $(M,g)$ be a compact Riemannian manifold of dimension $d$. Let $(U';\varphi =x^1,\cdots, x^d)$ be a chart of M, $U\subset\subset U'$ be an open set of $U'$. $A^p(M)$ denotes the set of smooth ...
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1answer
19 views

Which parts of $S$ these parametrization cover?

Show that the set $S=\{(x,y,z): z=x^2-y^2\}$ is a regular surface and check that parts a and b are parametrizations for $S$: a. $x(u,v)=(u+v, u-v, 4uv)$ with $(u, v)\in\mathbb{R}^2$ b. ...
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1answer
51 views

$(1,0)$-forms on a complex manifold

I'm reading in a paper: "Let $\theta$ be a $(1,0)$-form with respect to $I$ ($I \theta = -i \theta$)". Here $I$ is the almost complex structure. Any idea why it says $I \theta = -i \theta$ and not $I ...
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36 views

Proof of Euler's Theorem on differential geometry [on hold]

Proof of Euler's Theorem on Differential Geometry: K(A) = k1 Cos^(A) + k2 Sin^(A) Thank you.
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1answer
23 views

Tangent of evolute and singed curvature

This is an exercise from differential geometry textbook by Do Carmo. Let $\alpha:I\to \mathbb{R}^2$ be a regular parametrized plain curve (arbitrary parameter), define $n=n(t)$ and $k=k(t)$, where ...
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37 views

Notion of a distribution as acting on tangent spaces

I'm reading a paper that uses distributions in a way I'm not entirely comfortable with. To be precise, I'm not sure what definitions the author is working with and can't find any natural way to fill ...
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Gauss curvature K in polar coordinates

EDIT: A surface is given in Monge's form: $z=f (x,y)$ the partial derivatives of $z$ are.. $$ p = \frac{\partial z}{\partial x}, \; q = \frac{\partial z}{\partial y}, \; ...
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1answer
36 views

Is this orientation preserving or reversing?

I am confused about the definition of orientation on manifolds. Let $X=\{(x,y,0)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ and $Y=\{(x,y,1)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ be two one dimensional circles in ...
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0answers
21 views

Orientation of surfaces

From the book: Fixing a parametrization $x(u,v)$ of a neighborhood of a point $p$ of a regular surface $S$, we determine an orientation of the tangent plane $T_p (S)$, namely, the orientation of the ...
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0answers
17 views

Are the integrals over the open upper half plane and that over the closed one the same?

I am reading the part about manifold with boundary and Stokes theorems. I am so confused with the terminology used there. Stokes theorems says that if $M$ is a $n$-dimensional oriented smooth ...
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1answer
34 views

Is a sphere really a (differentiable) manifold?

I am a beginning student in Differential Geometry. According to what I understand, the charts: $$\sigma_+^z(x,y) = (x,y, \sqrt{1 - x^2 - y^2} )$$ $$\sigma_+^x(u,v) = (\sqrt{1 - u^2 - v^2},u,v )$$ ...
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1answer
40 views

Inverse mapping for a simple $\mathbb{R}^3$ surface given by $(\sin u, \sin 2u, v)$.

For a domain $U=\{\, (u,v) \in \mathbb{R}^2 \mid -\pi<u<\pi,\ 0<v<1 \,\}$ we have a mapping $X \colon U \to \mathbb{R}^3$ defined by $X(u,v) = (\sin u, \sin 2u, v)$. The resulting surface ...
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Is it possible to rectify two linearly independent vectors by the same diffeomorphism to the first two unit vectors in $\mathbb R^n$?

Suppose we are given two vector fields $V_1$ and $V_2$ defined on $\mathbb R^n$ such that the vectors $V_1(x)$ and $V_2(x)$ are linearly independent for each $x$. Is is possible to find a ...
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25 views

Question about the existence of the flow of a vector field in $\mathbb R^3$.

Let $V_r$ be a smooth vector field defined on a sphere of radius $r$ that is always tangential to the sphere on which it is defined. Define a vector field on $\mathbb R^3$ by declaring $V(x) = ...
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1answer
22 views

Symplectic form and volume of parallelepiped

Define the canonical symplectic form $\omega$ on $\mathbb{R}^{2n}$ by $\omega(u,v)=u^TJv$, where $$J=\begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}.$$ I do not understand why the volume ...
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0answers
28 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
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0answers
16 views

Geodesic formulation from surface parametrization

What differential relation f (u,v,du/dv)=0 can be used to convert parametrization of a two parameter surface X(u,v) into one parameter geodesics on surface X ?
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22 views

Evolute of a cycloid

Find the evolute of the cycloid: $x = u + \sin(u)$, $y = 1 + \cos(u)$ Hint: given is that $T = \big(\cos (u/2), − \sin(u/2)\big)$. Is it that we differentiate $x$ and $y$ and then make it ...
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1answer
49 views

the shortest path between two points and the unit sphere and the arc of the great circle

Prove that the shortest path between two points on the unit sphere is an arc of a great circle connecting them Great Circle: the equator or any circle obtained from the equator by rotating further: ...
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1answer
34 views

computing transition function of tangent bundle $S^n$

I'm just starting to learn about vector bundles, I want to compute the transition functions of the bundle $TS^n$. I started with the stereographic atlas $U_1 = S^n - \{N\}$ and $U_2 = S^n - \{S\}$ ...
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20 views

Uncertain about answers computing area and volume of sphere with unusual metric.

Consider the a metric in a three dimensional space given by $$ ds^{2} = \frac{dr^{2}}{1-\frac {2}{r}} + r^{2} (d\theta^{2} + sin^{2} (\theta) \: d\phi^{2}) $$ Calculate: a) The area of a sphere ...
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1answer
29 views

Riemannian geometry algebra

Is this derivation correct? $$ R^{ab}_{;a}=0 $$ $$ g_{ac}g_{bd}R^{ab}_{;a}=0 $$ $$ (g_{ac}g_{bd}R^{ab})_{;a}=0 $$ $$ R_{cd;a}=0 $$ And does that mean I now have $n^3$ equation as opposed to $n$?
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0answers
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Some problem similar to Dido's problem [duplicate]

The question is : "Let $A$ and $B$ be two fixed points in $\mathbb{R}^{2}$. Given $L>$ length of $AB$. Show that the curve $\alpha$ joining A and B, with length $L$, which together with AB forms a ...
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mean curvature of a nonlinear parabolic PDE

I want to compute the mean curvature of a reaction diffusion equations of the form $$ u_t=u_{xx}-u^2 $$ Any suggestions is appreciated!
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1answer
24 views

Area of pseudospherical segment

Surface area of segment of a sphere radius $a$ at the equator, between two parallels, is given by $ 2 \pi a (z_2-z_1) $,where $z_2, z_1$ are heights of spherical segment at radii of parallel circles ...
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1answer
33 views

Under what condition on f is this parametrized curve regular?

Consider a parametrized curve in $\mathbb R^2$ given by $$ \gamma (t)=(f(t)\cos(t), f(t)\sin(t)) $$ where $f$ is a smooth function of $t$. Under what condition on $f$ is $\gamma$ regular? I took the ...
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1answer
48 views

Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
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0answers
38 views

Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
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1answer
28 views

Question surface

We consider the surface $S$ that is defined as the graph of the function $z=2x^2-y^2, x,y \in \mathbb{R}$ Find a basis of the tangent plane $T$ of the surface $S$ at the point $M=(-1,2,-1)$ Find a ...
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1answer
35 views

Relation between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$?

This question is a little bit of a shot in the dark, but maybe someone stumbled over it before... Let $M$ be a (simply connected) smooth manifold modelled on a locally convex space $X$ over ...
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1answer
62 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
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1answer
50 views

Proof of curvature of a curve described by Polar Coordinates

I have been looking everywhere for a proof on the curvature of a plane curve that is represented in polar coordinates. I am close in proving it myself, however, I seem to be missing a particular part ...